2017 Putnam A3

Let $a$ and $b$ be real numbers with $a<b$, and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0, \infty)$

such that $\int_a^b f(x)\,dx = \int_a^b g(x)\,dx$ but $f \neq g$. For every positive integer $n$, define

$$

I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}\,dx.

$$

Show that $I_1, I_2, I_3, \dots$ is an increasing sequence with $\lim_{n \to \infty} I_n = \infty$.

设$a$和$b$为$a<b$的实数，并设$f$和$g$为从$[a,b]$到$(0, \infty)$的连续函数

使得 $\int_a^b f(x)\,dx = \int_a^b g(x)\,dx$ 但 $f \neq g$。对于每个正整数 $n$，定义

$$

I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}\,dx。

$$

证明 $I_1, I_2, I_3, \dots$ 是一个递增序列，$\lim_{n \to \infty} I_n = \infty$。